\begin{tabular}{|l|l} Divide using synthetic division. & Choose the correct quotient. \ $$ \frac{2 r^{3}+7 r^{2}+5 r-2}{r+2} $$ & A. $$ 2 r^{2}-3 r+1 $$ \ \hline & B. $$ 2 r^{2}-3 r-1 $$ \ \hline $$ 2 r^{2}+3 r-1 $$ \ \hline\end{tabular}

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To divide the polynomial $$ 2r^3 + 7r^2 + 5r - 2 $$ by $$ r + 2 $$ using synthetic division, we follow these steps:

1. **Set up the synthetic division**: We will use the root of the divisor $$ r + 2 = 0 $$, which gives us $$ r = -2 $$.

2. **Write down the coefficients**: The coefficients of the polynomial $$ 2r^3 + 7r^2 + 5r - 2 $$ are $$ 2, 7, 5, -2 $$.

3. **Perform synthetic division**:
   - Write the coefficients in a row.
   - Bring down the leading coefficient.
   - Multiply the root by the value just brought down and add it to the next coefficient.
   - Repeat this process until all coefficients have been processed.

Let's perform the synthetic division step by step:

- Coefficients: $$ 2, 7, 5, -2 $$
- Root: $$ -2 $$

$$
\begin{array}{r|rrrr}
-2 & 2 & 7 & 5 & -2 \
   &   & -4 & -6 & -2 \
\hline
   & 2 & 3 & -1 & 0 \
\end{array}
$$

4. **Interpret the result**: The bottom row gives us the coefficients of the quotient polynomial. The last number (0) is the remainder. The quotient is $$ 2r^2 + 3r - 1 $$.

Now, let's compare the quotient $$ 2r^2 + 3r - 1 $$ with the options provided:

- A. $$ 2r^2 - 3r + 1 $$
- B. $$ 2r^2 - 3r - 1 $$
- C. $$ 2r^2 + 3r - 1 $$

The correct quotient is **C. $$ 2r^2 + 3r - 1 $$**.
The correct quotient is $$ 2r^2 + 3r - 1 $$.

\begin{tabular}{|l|l} Divide using synthetic division. & Choose the correct quotient. \\ \( \frac{2 r^{3}+7 r^{2}+5 r-2}{r+2} \) & A. \( 2 r^{2}-3 r+1 \) \\ \hline & B. \( 2 r^{2}-3 r-1 \) \\ \hline \( 2 r^{2}+3 r-1 \) \\ \hline\end{tabular}

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